In a bowl combine the tuna, scallions, gingerroot, mustard, soy sauce, cayenne, salt and pepper. Form the mixture into six 1-inch patties and chill for 30 minutes.
In a small bowl whisk together the mayonnaise, wasabi, and lime juice. Cover and chill.
In a medium bowl combine the cucumber, red onion, pickled ginger, and rice vinegar. Toss to coat and season with salt and pepper.
Heat a dry large heavy skillet over moderately high heat. Brush each tuna pattie with the vegetable oil. Cook the tuna patties for 5 to 6 minutes per side until they are just cooked through. Transfer the tuna patties to the buns and top with the wasabi mayonnaise and the pickled cucumber ginger relish.